algebrafractions

laws of exponents

add/sub: find a common denominatormultiply: straight acrossdivide: reciprocate bottom, then multiply

exponents• #59 =

83!

• #1#> = #14>

• 3"

3# = #15>

radicals• !

!@ = ! ! @ = ! ⁄$ !

• EVEN ROOTS• !

!9 = !• don’t need abs values if

power outside root is even• ODD ROOTS

• !!9 = !

intervals & sets interval number line inequality

( or ) open circle < or >

[ or ] closed circle ≤ or ≥

• union (∪): combination of both sets• intersection (∩): overlap between sets

functionsa function passes the vertical line test

lines• slope: H =

?BCD?E9

=>%5>&1%51&

• ⊥ slope: HF = −8@

• point-slope form: 5 − 58 = H(! − !8)

piecewise functions

$ ! = J$K4LMNO4 1, POH#N4 1

$K4LMNO4 2, POH#N4 2

• graph each function, but erase anything that isn’t included in its associated domain

function symmetryEVEN ODD

symmetric about origin symmetric about 5-axis

$ −! = $(!) $ −! = −$(!)

function translationVertical Shift• UP: add a number outside function• DOWN: subtract a number outside functionHorizontal Shift• RIGHT: subtract a positive number inside function (see a negative)• LEFT: subtract a negative number inside function (see a positive)Reflections• x-axis: −$(!)• y-axis: $(−!)

changing form of a function

solving equations

system of equationsgraphically: where equations intersectalgebraically: substitution

1. solve one equation for one variable

2. sub equation 1 into equation 2 to solve for one variable

3. solve for the remaining variable

• common factor: something all terms share• special formulas

• !: − 5: = (! + 5)(! − 5)• (! + 5):= !: + 2!5 + 5:

• (! − 5):= !: − 2!5 + 5:

• !G + 5G = ! + 5 !: − !5 + 5:

• !G − 5G = ! − 5 !: + !5 + 5:

• standard factoring• factoring by grouping

• group terms with common factors• remove greatest common factor from

each group• end up with 2 factors

• long polynomial division• : is a root ⟺ ! − : is a factor• divide polynomial by known factor

factoring completing the squarechanges #!: + ;! + L to # ! − ℎ : + S

1. factor out # from all terms2. look at linear coefficient, divide by 2,

square it3. add and subtract this number4. identify the perfect square and combine

remaining numbers5. factor # back in

conjugates• the conjugate of # ± ; is # ∓ ;• multiply the top and bottom of a fraction

by its conjugate to simplify• (#1)>= #1>

• #1;1 = (#;)1

• 3"

=" =3=

1

linear equations• move all terms with variable to one side• factor variable if needed • isolate variable

quadratic equations• make sure quadratic is set equal

to 0 first! • factor, use quadratic formula, or

complete the square

other types of equations• zero factor property• isolate variable• cross multiply• find common denominator

pre-calcangles

• ⁄8 : circle = 180° = Y radians • radians à degrees: multiply by 8HI

J• degrees à radians: multiply by J

8HI

unit circle(Z[\ ] , \^_])

trig functionsinput = angleoutput = number sin a = #

sin acsc a =

1

sin a

cos asec a =

1

cos a

tan acot a =

1

tan a

EVEN functions• cos a• sec a

• sin a• csc a

trig function symmetry

• tan a

• cot a

ODD functions

evaluating trig functions

sin / csc /

cos / sec /

tan / cot /

SOHCAHTOAa

hypotenuse

adjacent

oppo

site

translating trig functionsVertical Shift• UP: add a number outside function• DOWN: subtract a number outside functionHorizontal Shift (phase shift)• RIGHT: subtract a positive number inside function (see -)• LEFT: subtract a negative number inside function (see +)Amplitude (how tall wave is)• multiplier in front of trig function; g sin(a)Period (how often wave repeats)• multiplier inside trig functions; sin(ha)• new period: K?BLB93M ND?BKO

P

period = 2Y period = 2Y

period = 2Y period = 2Y

period = Y period = Y

trig identitiescos& 8 + sin& 8 = 1

Limitslim#→% ) ! = + ⟺ lim#→%! ) ! = + = lim

#→%")(!)

computing limitsdirect sub

0 in denominator?

done! 0 in numerator?

• factor• conjugate• common denominators• split absolute value• expand• Squeeze Theorem

infinite limit

limits graphically

lim!→#$& ' = 5

lim!→#%& ' = 1

& −2 = −1

lim!→&! & ' = ∞

lim!→'" & ' = 4

lim!→(! & ' = 6

lim!→(" & ' = 2lim!→)& ' = 3

Squeeze Theoreminfinite limits

limits at infinity

If $ ! ≤ & ! ≤ ℎ ! and lim1→3 $ ! = lim1→3 ℎ ! = ,, then lim1→3 & ! = ,.

1. Begin with an inequality, $ ! ≤ & ! ≤ ℎ !(&(!) is the function you want to figure out the limit of)

2. Take the limit of the outer two functions.3. If the limits of the outer two functions evaluate to the same

number, the limit of the middle function also evaluates to the same number.

lim)→. * & = ∞

lim)→±** & = +

• look at left and right sided limits• find what is causing 0 in denominator• replace with small + or small –• 4

4 = ∞, 55 = ∞, 45 = −∞, 54 = −∞

1. Find the possible VA by solving denominator = 02. Verify ! = # is a VA with at least one infinite limit

vertical asymptotes

• lim1→±781! = 0

• 4 is even, lim1→±7 !9 = ∞

• 4 is odd, lim1→7 !9 = ∞ , lim1→57 !

9 = −∞

horizontal asymptotes5 = , is a HA if lim1→7 $ ! = , and/or lim1→57 $ ! = ,

slant asymptotesdegree of numerator = degree of denominator + 1long polynomial division

1. choose highest power in denominator2. divide each term by highest power3. simplify each term4. evaluate each term

computing limits at infinity ! →∞: !: = ! = !! → −∞: !: = ! = −!

need negative sign when ! → −∞ and you have an

odd exponent outside radical!

caution!

computing infinite limits

NO

NO

YES

YES 00

#0

inversesexponential functions logarithmic functions

function composition

inverse functions# $ = &!/ > 1 and / ≠ 1

laws of exponents

• ###$ = ##"$

• %"%# = ##!$

• ## $ = ##$

• #/ # = ##/#

• %&#= %"

&"

• ## = #$ ⇔ ! = 1

domain: (−∞,∞) range: (0,∞)

natural exponential2 ! = 3#3 ≈ 2.7183

# $ = log" $range: (−∞,∞)domain: (0,∞)

laws of logs• log= !5 = log= ! + log=(5)• log=

1>= log=(!) − log=(5)

• log= !? = : log=(!)• log= ; = 1

compute logslog= ! = 5 ⇔ ;> = !

“What power do I need to raise my base to in order to get !?”

natural log2 ! = ln(!)

basee

all laws of logs also apply to ln ! !

• a function has an inverse if it is one-to-one(passes horizontal line test)

• $(!) and $58(!) reflect over the line ? = @• $(!) and $58(!) swap domains and ranges• $ $58 ! = ! and $58 $ ! = !

properties of inverses

finding an inverse1. solve for !2. interchange ! and 53. replace 5 with inverse notation, $58(!)

&#$%A(!) = $

log"(&!) = $

B(C @ ): &(!) is the inner function and $(!) is the outer function

derivativescontinuity &

differentiabilitycontinuity conditions• $(#) defined• lim

1→3$ ! exists

• $ # = lim1→3

$ !

discontinuities• removable• jump• infinite• oscillating

Intermediate Value TheoremIf $ is continuous on #, ; and $(#) < , < $(;),

then there is a number L ∈ (#, ;) such that $ L = ,

differentiability

a function will not be differentiable at a point if there is a …differentiable = able to find a derivative

• discontinuity• corner• cusp• vertical tangent

differentiability ⇒ continuitycontinuity ⇏ differentiability (i.e. corner or cusp)

the derivativeis the slope

of thetangent line

derivative at a point

derivative as a function

2' # = lim#→%2 ! − 2(#)

! − # 2' # = lim)→*2 # + ℎ − 2(#)

ℎ

the slope of the tangent line at a single point, ! = #

the slope of the tangent line anywhere on the original function

2' ! = lim)→*2 ! + ℎ − 2(!)

ℎgiven $′(!), you can find the slope of the tangent line

anywhere you’d like by plugging in a value for !

derivative graphsA(?) A′(?)

increasing above !-axis

decreasing below !-axis

smooth min/max crosses !-axis

constant zero (over interval)

linear constant (slope of line)

quadratic linear

derivative rules• O

O1L = 0

• OO1

! = 1

• OO1

!9 = 4!958

• OO1

L$(!) = L$′(!)

• OO1

$ ! + &(!) = $Q ! + &′(!)

• OO1

m1 = m1

• product rule: OO1

$ ! & ! = $Q ! & ! + $ ! &Q(!)

• quotient rule: OO1

R(1)L(1)

=L 1 R' 1 5R 1 LQ(1)

(L 1 )%

• OO1(sin !) = cos !

• OO1

tan ! = sec: !

• OO1

sec ! = sec ! tan !

• OO1

cos ! = − sin !

• OO1

cot ! = −csc: !

• OO1

csc ! = − csc ! cot !

• chain rule: OO1

$ &(!) = $Q &(!) n &′(!)

derivative applicationsphysics• position: o(M)• velocity: p M = oQ M• acceleration: # M = pQ M = o′′(M)

MAX HEIGHT: when v M = 0; plug time into o(M)VELOCITY ON GROUND: when o M = 0; plug time into p(M)SPEED: speed = p(M)SLOWING DOWN: p(M) and #(M) opposite signs SPPEDING UP: p(M) and #(M) same signs